Estimating Square Roots

Mental Math · Axiom Academy

Mental techniques for approximating square roots without a calculator The foundation of estimating square roots is knowing your perfect squares. By identifying which two perfect squares your target number falls between, you can immediately establish bounds for your estimate. Strategy: To estimate √50, recognize that 49 < 50 < 64, so 7 < √50 < 8. This tells us √50 is between 7 and 8. 2. Linear Interpolation Method Once you've identified the bounding perfect squares, use linear interpolation to refine your estimate. This technique assumes the square root function grows approximately linearly between perfect squares. 50 is 1 unit above 49, out of a total gap of 15 units (64 - 49) Actual: √50 ≈ 7.071 (very close!) 3. Babylonian Refinement (Newton's Method) For even greater accuracy, use the ancient Babylonian method: take your initial estimate, divide the target number by it, then average the result with your estimate. One iteration dramatically improves accuracy! This matches the actual value √50 ≈ 7.071! For rapid mental estimation without calculations, use these practical shortcuts that work well for most everyday situations. √20: Between 4² and 5² (16 and 25). Closer to 16, so ≈ 4.5 (actual: 4.47) √75: Between 8² and 9² (64 and 81). Closer to 81, so ≈ 8.7 (actual: 8.66) √130: Between 11² and 12² (121 and 144). Middle-ish, so ≈ 11.4 (actual: 11.40) √200: Between 14² and 15² (196 and 225). Close to 196, so ≈ 14.1 (actual: 14.14)

This is the written version of the interactive lesson above. See the full Mental Math course.